In Exercises 9–24, solve each triangle. Round lengths to the near...

Question

In Exercises 9–24, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.a = 5, c = 2, B = 90°

Accepted Answer

Hello everybody. I hope you're doing. All right. Today, today, we're going to be looking at this question that states find the missing side length and angles of the triangle and express the length to one decimal place and the angle to the nearest degree. Now, they tell us a couple of details about our triangle. Now, we know that sides will be noted by a lower case letter and angles will be noted by a upper case letter. So we have that side P equals nine and side R equals four. While angle Q will be equal to 90 degrees. Now, s choice A says that angle P is equal to 24 degrees. Side Q is equal to 9.8 units and angle R is equal to 66 degrees. Now, for all the S choices P will be an angle and R will be an angle while Q is a side, as choice B says that P equals 66 degrees, Q equals 9.8 units and R is degrees. Answer choice C says that P is 66 degrees, Q is 10 units and R is 24 degrees. And as for choice D says that P is 24 degrees Q equals 10 units and R is 66 degrees. Now, the first thing we can do once I me this question is trying to draw a picture of our triangle. So let's say we have, we know that we have a 90 degree angle. So we're going to have a right triangle. So accordingly, we draw a right triangle. Now that right angle, the 90 degree angle will be angle Q and we know that the side opposite of angle Q will be side Q because angle Q opens up towards its corresponding side side Q. Now, as you'll see as we move forward, how we label the rest isn't necessarily important as long as we label the angles corresponding to the correct side. So we have angle R opening up to side R and angle P opening up to side P. Now we have a right triangle. And when dealing with a right triangle, we have, we have something called Pythagorean theorem and this will help us solve for the missing side. Now, Python theorem states that A squared plus B squared equals C squared where A and B are the legs of the right triangle and C is the hypos. So in our case, this will be P squared plus R squared equals Q squared. And we can go ahead and plug in our values to have nine squared plus four squared equals Q squared. So nine squared is 81 plus four squared, which is 16, that'll be equal to Q squared. So we'll have that 97 equals Q squared. And we just take the square root of both sides to get rid of the square. And we'll have the, the square root of 97 is equal to Q, which once we plug that into our calculator, we have that queue is around 9.84 and so on. But we have to cut it off at one decimal place as they asked us in the question. So have that Q is equal is around 9.8. And that will be our first answer for this question. Now, another thing that we know for a right triangle, we know that in a right triangle cosine of theta is equal to adjacent divided by hypos. And what do I mean by this? Well, let's give an example. So let's say we're trying to solve for angle P and we have cosine of angle P. This will be equal to the adjacent side divided by the hypo side. The adjacent side of angle P is R and this will be divided by the hypotenuse Q. So we'll have that cosine of P is equal to four divided by 9.8, which one simplified will be close on A P is equal to 0.4081 and solved for the angle will do what we refer to as inverse cosine. So we'll have inverse cosine denoted by the negative one exponent. And we'll have inverse cosine of the value we just calculated. So 0.4081 that will be equal to the angle P. So once we plug that into our calculator, we'll come out with P angle P is around 65.91 degrees which will round off to be 66 degrees. And that will be our second answer and our answer for angle P. Now, finally, for the last angle, there's another thing that we can note and that is that a triangle is equal to 180 degrees. Therefore, angle P plus angle, Q plus angle R are equal to 180 degrees. So when you add up the, the angles of a triangle, they would, they will add up to 180. So we'll have that 66 degrees plus degrees plus R is equal to degrees, which means that R is equal to 180 degrees minus 66 degrees minus 90 degrees, which means that R is equal to 24 degrees. And that will be our final answer. And we can see that this matches with answer choice B where P angle P is 66 degrees, side Q is 9.8 units and angle R is degrees. So you can see that it's very important to know the different rules that correspond to the specific type of triangle that we're working with. So I hope that was helpful and until next time.

In Exercises 9–24, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.a = 5, c = 2, B = 90°

Key Concepts

Law of Sines

Right Triangle Properties

Pythagorean Theorem

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